It should be noted that the above analysis actually used weighted versions of
the Y vectors to include a soft memory decay
process. Figure 4.4 depicts the pre-processing,
exponential weighting and dimensionality reduction. Instead of
abruptly cutting off the Y at the T'th sample of ,
a
smooth decay is applied to more distant observations.

Figure 4.4:
Exponential Decay and Pre-Processing

Recall that the large vectors Y(t) represent a full window of past
interaction (short term memory). This window effectively covers
seconds of temporal data. This data is weighted
with an exponential decay which scales down
vectors that
constitute the big Y(t) vector. The further back in time a component is, the more its amplitude is attenuated. Thus, an
exponential ramp function is multiplied with each Y window (i.e. a
few seconds of each of the 30 time series). This reflects our
intuition that the more temporally distant the elements in the time
series, the less relevant they are for prediction. This decay agrees
with some aspects of cognitive models obtained from psychological
studies [16]. Once the vectors have been attenuated,
they form a new 'exponentially decayed' short term memory window
.
The process is shown in Figure 4.4
where a window is placed over the time series, generating a short term
memory Y. An exponential decay function is used to decay it and
generates the
version. The eigenspace previously discussed
is really formed over the
distribution and representing
in only this subspace (i.e. the top eigenvectors) generates
the compact vector
.
This is the final, low dimensional
representation of the gestural interaction between the two humans over
the past few seconds.